We show that the Dirichlet to Neumann map for the equation ∇·σ∇u = 0 in a two-dimensional domain uniquely determines the bounded measurable conductivity σ. This gives a positive answer to a question of A. P. Calderón from 1980. Earlier the result has been shown only for conductivities that are sufficiently smooth. In higher dimensions the problem remains open.

We study time harmonic scattering for the Helmholtz equation in R n . We show that certain penetrable scatterers with rectangular corners scatter every incident wave nontrivially. Even though these scatterers have interior transmission eigenvalues, the relative scattering (a.k.a. far field) operator has a trivial kernel and cokernel at every real wavenumber. IntroductionThe diffraction of light around corners and edges, and through slits, provided the first evidence for the wave nature of light. The diffraction patterns caused by plane waves incident on corners or edges and were among the first scattered waves to be calculated [14]. Geometric optics expansions for scattered waves [9] reveal the presence of scattered waves in regions where the simple theory of optics does not. Much of our understanding of classical electromagnetism is based on these patterns. This is why a stealth airplane is built to minimize the scattering from corners and edges.Although the single frequency inverse scattering problem has a unique solution, the wave scattered from a single incident wave does not contain enough information to determine an obstacle or a penetrable scatterer. In many cases, the same scattered wave might have been scattered by a scatterer supported on a smaller set. In this paper we will show that, a penetrable scatterer whose support contains a right angle corner as an extreme point of its convex hull

The interior transmission problem is a boundary value problem that plays a basic role in inverse scattering theory but unfortunately does not seem to be included in any existing theory in partial differential equations.This paper presents old and new results for the interior transmission problem ,in particular its relation to inverse scattering theory and new results on the spectral theory associated with this class of boundary value problems.

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Abstract. The scattering of a time-harmonic plane wave in an inhomogeneous medium is modeled by the scattering problem for the Helmholtz equation. A transmission eigenvalue is a wavenumber at which the scattering operator has a non-trivial kernel or cokernel. Because many sampling methods for locating scatterers succeed only at wavenumbers that are not transmission eigenvalues, they have been studied for some time. Nevertheless, the existence of transmission eigenvalues has previously been proved only for radial scatterers. In this paper, we prove existence for scatterers without radial symmetry.

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Abstract. We prove the absence of non-scattering energies for potentials in the plane having a corner of angle smaller than π. This extends the earlier result of Blåsten, Päivärinta and Sylvester who considered rectangular corners. In three dimensions, we prove a similar result for any potential with a circular conic corner whose opening angle is outside a countable subset of (0, π).

We prove the existence of complex geometrical optics solutions for Lipschitz conductivities. Moreover we show that, in dimensions n ≥ 3 that one can uniquely recover a W 3/2,∞ conductivity from its associated Dirichlet-to-Neumann map or voltage to current map.